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Theorem iunxun 3709
 Description: Separate a union in the index of an indexed union. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
Assertion
Ref Expression
iunxun x (AB)𝐶 = ( x A 𝐶 x B 𝐶)

Proof of Theorem iunxun
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 rexun 3100 . . . 4 (x (AB)y 𝐶 ↔ (x A y 𝐶 x B y 𝐶))
2 eliun 3635 . . . . 5 (y x A 𝐶x A y 𝐶)
3 eliun 3635 . . . . 5 (y x B 𝐶x B y 𝐶)
42, 3orbi12i 668 . . . 4 ((y x A 𝐶 y x B 𝐶) ↔ (x A y 𝐶 x B y 𝐶))
51, 4bitr4i 176 . . 3 (x (AB)y 𝐶 ↔ (y x A 𝐶 y x B 𝐶))
6 eliun 3635 . . 3 (y x (AB)𝐶x (AB)y 𝐶)
7 elun 3061 . . 3 (y ( x A 𝐶 x B 𝐶) ↔ (y x A 𝐶 y x B 𝐶))
85, 6, 73bitr4i 201 . 2 (y x (AB)𝐶y ( x A 𝐶 x B 𝐶))
98eqriv 2019 1 x (AB)𝐶 = ( x A 𝐶 x B 𝐶)
 Colors of variables: wff set class Syntax hints:   ∨ wo 616   = wceq 1228   ∈ wcel 1374  ∃wrex 2285   ∪ cun 2892  ∪ ciun 3631 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-un 2899  df-iun 3633 This theorem is referenced by:  iunsuc  4106  rdgisuc1  5891  oasuc  5959  omsuc  5966
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